# Re: Paper and slides on indefiniteness of CH

Dear Peter,

I think we should all be grateful to you for this eloquent description of how we gather evidence for new axioms based on the development of set theory. The first two examples (and possibly the third) that you present are beautiful cases of how a body of ideas converges on the formulation of a principle or principles with great explanatory power for topics which lie at the heart of the subject. Surely we have to congratulate those who have facilitated the results on determinacy and forcing axioms (and perhaps in time Hugh for his work on Ultimate L) for making this possible. Further, the examples mentioned meet your high standard for any such programme, which is that it “makes predictions which are later verified”.

I cannot imagine a more powerful statement of how Type 1 evidence for the truth of new axioms works, where again by “Type 1″ I refer to set theory’s role as a field of mathematics and therefore by “Type 1 evidence” I mean evidence for the truth of a new axiom based on its importance for generating “good set theory”, in the sense that Pen has repeatedly emphasized.

But I do think that what you present is only part of the picture. Set theory is surely a field of mathematics that has its own key questions and as it evolves new ideas are introduced which clarify those questions. But surely other areas of mathematics share that feature, even if they are free of questions of independence; they can have analogous debates about which developments are most important for the field, just as in set theory. So what you describe could be analagously described in other areas of mathematics, where “predictions” are made about how certain approaches will lead to the solution of central open problems. Briefly put: In your description of programmes for set theory, you treat set theory in the same way as one would treat any field of mathematics.

But set theory is much more that. Before I discuss this key point, let me interrupt myself with a brief reference to where this whole e-mail thread began, Sol’s comments about the indefiniteness of CH. As I have emphasized, there is no evidence that the pursuit of programmes like the ones you describe will agree on CH. Look at your 3 examples: The first has no opinion on CH, the second denies it and the third confirms it! I see set theory as a rich and developing subject, constantly transforming itself with new ideas, and as a result of that I think it unreasonable based on past and current evidence to think that CH will be decided by the Type 1 evidence that you describe. Pen’s suggestion that perhaps there will be a theory “whose virtues swamp the rest” is wishful thinking. Thus if we take only Type 1 evidence for the truth of new axioms into account (Sol rightly pointed out the misuse of the term “axiom” and Shelah rightly suggested the better term “semi-axiom”), we will not resolve CH and I expect that we won’t resolve much at all. Something more is needed if your goal is to say something about truth in set theory. (Of coures it is fine to not have that goal, and only a handful of set-theorists have that goal.)

OK, back to the point that set theory is more than just a branch of mathematics. Set theory also has a role as a foundation for mathematics (Type 2). Can we really assume that Type 1 axioms like the ones you suggest in your three examples are the optimal ones for the role of set theory as a foundation? Do we really have a clear understanding of what axioms are optimal in this sense? I think it is clear that we do not.

The preliminary evidence would suggest that of the three examples you mention, the first and third are quite irrelevant to mathematics outside of set theory and the second (Forcing Axioms) is of great value to mathematics outside of set theory. Should we really ignore this in a discussion of set-theoretic truth? I mean set theory is a great branch of mathematics, rife with ideas, but can we really assert the “truth” of an axiom which serves set theory’s needs when other axioms that contradict it do a better job in providing other areas of mathematics what they need?

There is even more to the picture, beyond set theory as a branch of or a foundation for math. I am referring to its Type 3 role, as a study of the concept of set. There is widespread agreement that this concept entails the maximality of V in height and width. The challenge is to explain this feature in mathematical terms, the goal of the HP. There is no a priori reason whatsoever to assume that the mathematical consequences of maximality in this sense will conform to axioms which best serve the Type 1 or Type 2 needs of set theory (as a branch of or foundation for mathematics). Moreover, to pursue this programme requires a very different approach than what is familiar to the Type 1 set-theorist, perfectly described in your previous e-mail. I am asking you to please be open-minded about this, because the standards you set and the assumptions that you make when pursuing new axioms for “good set theory” do not apply when pursuing consequences of maximality in the HP. The HP is a very different kind of programme.

To illustrate this, let me begin with two quotes which illustrate the difference and set the tone for the HP:

I said to Hugh:

The basic problem with what you are saying is that you are letting set-theoretic practice dictate the investigation of set-theoretic truth!

In other words, my starting point is not what facilitates the “best set theory”, but what one can understand about maximality of V in height and width.

On a recent occasion, Hugh said to me:

[Yet] you propose to deduce the non existence of large cardinals at some level based on maximality considerations. I would do the reverse, revise maximality.

This second quote precisely indicates the difference in our points of view. The HP is intended to be an unbiased analysis of the maximality of V in height and width, grounded in our intuitions about this feature and limited by what is possible mathematically. These intuitions are indeed fairly robust, surely more so than our judgments about what is “good set theory”. I know of no persuasive argument that large cardinal existence (beyond what is compatible with V = L) follows from the maximlity of V in height and width. Indeed in the literature authors such as Gödel had doubts about this, whereas they have felt that inaccessible cardinals are derivable from maximality in height.

So the only reasonable interpretation of Hugh’s comment is that he feels that LC existence is necessary for “good set theory” and that such Type 1 evidence should override any investigation of the maximality of V in height and width. Pen and I discussed this (in what seems like) ages ago in the terminology of “veto power” and I came to the conclusion that it should not be the intention of the HP to have its choice of criteria dictated by what is good for the practice of set theory as mathematics.

To repeat, the HP works like this: We have an intuition about maximality (of V in height and width) which we can test out with various criteria. It is a lengthy process by which we formulate, investigate and compare different criteria. Sometimes we “unify” or “synthesise” two criteria into one, resulting in a new criterion that based on our intuitions about maximality does a better job of expressing this feature than did the individual criteria which were unified. And sometimes our criteria conflict with reality, namely they are shown to be inconsistent in ZFC. Here are some examples:

Synthesis: The IMH is the most obvious criterion for expressing the maximality of V in width. #-generation is the strongest criterion for expressing the maximality of V in height. If we unify these we get IMH#, which is consistent but behaves differently than either the IMH alone or #-generation alone. Our intuition says that the IMH# better expresses maximality than either the IMH alone or #-generation alone.

Inconsistency (examples with HOD): We can consistently assert the maximality principle $V \noteq \text{HOD}$. A natural strengthening is that $\alpha^+$ of HOD is less than $\alpha^+$ for all infinite cardinals $\alpha$. Still consistent. But then we go to the further natural strengthening $\alpha^+$ of $HOD_x$ is less than $\alpha^+$ for all subsets $x$ of $\alpha$ (for all infinite cardinals $\alpha$). This is inconsistent. So we back off to the latter but only for $\alpha$ of cofinality $\omega$. Now it is consistent for many such $\alpha$, not yet known to be consistent for all such $\alpha$. We continue to explore the limits of maximality in this way, in light of what is consistent with ZFC. A similar issue arises with the statement that $\alpha$ is inaccessible in HOD for all infinite regular $\alpha$, which is not yet known to be consistent (my belief is that it is).

The process continues. There is a constant interplay betrween criteria suggested by our maximality intuitions and the mathematics behind these criteria. Obviously we have to modify what we are doing as we learn more of the mathematics. Indeed, as you pointed out in your more recent e-mail, there are maximality criteria which contradict ZFC; this has been obvious for a long time, in light of Vopenka’s theorem.

It may be too much to ask that your program at this stage make such predictions. But I hope that it aspires to that. For if it does not then, as I mentioned earlier, one has the suspicion that it is infinitely revisable and “not even wrong”.

Once again, the aim of the programme is to understand the consequences of the maximality of V in height and width. Your criterion of “making predictions” may be fine for your Type 1 programmes, which are grounded by nothing more than “good set theory”, but it is not appropriate for the HP. That is because the HP is grounded by an intrinsic feature of the set-concept, maximality, which will take a long time to understand. I see no basis for your suggestion that the programme is “infinitely revisable”, it simply requires a huge amount of mathematics to carry out. Already the synthesis of the IMH with #-generation is considerable progress, although to get a deeper understanding we’ll definitely have to deal with the $\textsf{SIMH}^\#$ and HOD-maximality.

If you insist on a “prediction” the best I can do is to say that the way things look now, at this very preliminary stage of the programme, I would guess that both not-CH and the nonexistence of supercompacts will come out. But that can’t be more than a guess at this point.

Now I ask you this: Suppose we have two Type 1 axioms, like the ones in your examples. Suppose that one is better than the other for Type 2 reasons, i.e., is more effective for mathematics outside of set theory. Does that tip the balance between those two Type 1 axioms in terms of which is closer to the truth? And I ask the same question for Type 3: Could you imagine joining forces and giving priority to axioms that both serve the needs of set theory as mathematics and are derivable from the maximality of V in height and width?

One additional worry is the vagueness of the idea of the ” ‘maximal’ iterative conception of set”. If there were a lot of convergence in what was being mined from this concept then one might think that it was clear after all. But I have not seen a lot of convergence. Moreover, while you first claimed to be getting “intrinsic justifications” (an epistemically secure sort of thing) now you are claiming to arrive only at “intrinsic heuristics” (a rather loose sort of thing). To be sure, a vague notion can lead to motivations that lead to a great deal of wonderful and intriguing mathematics. And this has clearly happened in your work. But to get more than interesting mathematical results — to make a case for for new axioms — at some stage one will have to do more than generate suggestions — one will have to start producing propositions which if proved would support the program and if refuted would weaken the program.

I imagine you agree and that that is the position that you ultimately want to be in.

No, the demands you want to make of a programme are appropriate for finding the right axioms for “good set theory” but not for an analysis of the maximality of V in height and width. For the latter it is more than sufficient to analyse the natural candidates for maximality criteria provided by our intuitions and achieve a synthesis. I predict that this will happen with striking consequences, but those consequences cannot be predicted without a lot of hard work.

Thanks,
Sy

PS: The above also addresses your more recent mail: I don’t reject a form of maximality just because it contradicts supercompacts (because I don’t see how supercompact existence is derivable form any form of maximality) and I don’t see any problem with rejecting maximality principles that contradict ZFC, simply because by convention ZFC is taken in the HP as the standard theory.

PPS: A somewhat weird but possibly interesting investigation would indeed be to drop the ZFC convention and examine criteria for the maximality of V in height and width over a weaker theory.

# Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I’d like to add one more general observation to these [Koellner's] compelling remarks. In cases like this one, where the evidence is overwhelmingly extrinsic, there’s no reason to expect the relevant discussions to be ‘generally understandable’. It’s only in cases like the HP, where the evidence is intended to be primarily or even exclusively intrinsic, that ‘general understandability’ becomes crucial. (If the principles are supposed to follow from the concept of set, then we need to be able to see how that works.)

All best,

Pen

PS to Peter: Thank you very much for your message and slides about the current state of understanding on choiceless cardinals! I look forward to spending a bit more time with them both.

# Re: Paper and slides on indefiniteness of CH

Dear Pen and Sy,

I have benefited from your exchange. I’ll try to add some input.

Sy: I have wanted to say something about your proposal. But I am still very unclear on how you understand the philosophical landscape, in particular, on how you understand the distinction between intrinsic and extrinsic justification.

One of the distinctive aspects of your view — a selling point, from a philosophical point of view — is that it involves pursuit of “intrinsic justifications”, as opposed to “extrinsic justifications”. But I am not sure how you are using these terms. From your exchange with Pen it seems that your usage is quite different from the original, namely, that of Gödel.

For Gödel a statements S is intrinsically justified relative to a concept C (like the concept of set) if it “follows from” (or it “unfolds”, or is “part of”) that concept. The precise concept intended is far from clear but it seems clear that whatever it is intrinsic justifications are supposed to be very secure, not easily open to revision, and qualify as analytic. In contrast, on your usage it appears that intrinsic justifications need not be secure, are easily open to revision, and (so) are (probably) not analytic.

For Gödel a statement S is “extrinsically justified” relative to a concept C (like the concept of set) if it is justified (on the basis of reasons grounded in that concept) in terms of its consequences (especially its “verifiable” consequences), just as in physics. Again this is far from precise but it seems clear that extrinsic justifications are not as secure as intrinsic justifications but instead offer “probable”, defeasible evidence. In contrast, on your usage it appears that you do not understand “extrinsic justification” as an epistemic notion, but rather you understand it as a practical notion, one having to do with meeting the aims of a pre-established practice.

So, you appear to use “intrinsic justification” for an epistemic notion that is not as secure as the traditional notion but rather merely gives epistemic weight that falls short of being conclusive. Moreover, at points, when talking about intrinsic justifications you talk of testing them in terms of their consequences. So I think that by “intrinsically justified” you mean either “intrinsically plausible” or “extrinsically justified”.

I think you need to be more precise about how you use these terms and how your usage relates to the standard usage. This is especially important if the main philosophical selling point of your proposal is that it is re-invigorating “intrinsic justifications” in the sense of Gödel. (Good places to start in getting clear on these notions are the papers of Tony Martin and Charles Parsons.)

In what I say next I will use “intrinsic justification” in the standard sense, both for the sake of definiteness and because it is on this understanding that your view is distinctive from a philosophical point of view.

Let me begin with a qualification. I am generally wary of appeals to
“intrinsic justification”, for the same reason I am generally wary of
appeals to “self-evidence”, the reason being that in each case the
notion is too absolute — it pretends to be a final certificate, an
epistemic high-ground, a final court of appeal. But in fact there is
little agreement on what is intrinsically justified (and on what is
self-evident). For this reason, in the end, discussions that employ
these notions tend to degenerate into foot-stamping. It is much
better, I think, to employ notions where there is widespread
intersubjective agreement, such as the relativized versions of these notions, notions like “A is more intrinsically plausible than B” and “A is more (intrinsically) evident than B”. This is one reason I find
extrinsic justifications to be more helpful. They are piecemeal and
modest and open to revision under systematic investigation. (I think
you agree, since I think that ultimately by “intrinsic justification”
you mean what is normally meant by “extrinsic justification”).

But let me set that qualification aside and proceed, employing the notion of “intrinsic justification” in the standard sense, for the
reasons given above.

There is an initial puzzle that arises with your view.

THE PUZZLE

1. You claim that IMH is intrinsically justified.
2. You claim that inaccessible cardinals — and much more — are intrinsically justified
3. FACT: IMH is implies there are no inaccessibles.

The natural reaction at this point would be to think that there is

But perhaps there is a subtlety. Perhaps in (1) and (2) intrinsic
justifications are relative to different conceptions.

When you claim that IMH is intrinsically justified what exactly are you saying and what is the case for the claim? Are you saying IMH (a) intrinsically justified relative to our concept of set (which, on the face of it, concerns V) or (b) the concept of being a countable transitive model of ZFC, or (c) the concept of being a countable transitive model of ZFC that meets certain other constraints? Let’s go through these options one by one.

(a) IMH is intrinsically justified relative to the concept of set. I don’t see the basis for this claim. To the extent that I have a grasp on the notion of being intrinsically justified relative to the concept of set I can go along with the claims that Extensionality and Foundation are so justified and even the claims that Infinity and Replacement and Inaccessibles are so justified (thus following Gödel and others) but I lose grip when it comes to IMH. Moreover, IMH implies that there are no inaccessibles. Does that not undermine the claim that IMH is intrinsically justified on the basis of the concept of set? Assuming it does (and that this is not what you claim) let’s move on.

(b) IMH is intrinsically justified relative to the concept of being a
countable transitive model of ZFC. I have a good grasp on the notion of being a countable transitive model of ZFC. And I think it is interesting to study this space. But when I reflection this space –when I try to unfold the content implicit in this idea — I can reach nothing like IMH.

(c) IMH is intrinsically justified relative to the concept of being a countable transitive model of ZFC that meets certain other constraints. I can certainly see going along with this. But, of course, it depends on what the other constraints are. We have two options: (i) We can be precise about what we mean. For example, we can build into the notion that we are talking about the concept of being a countable transitive model of ZFC that satisfies X, where X is
something precise. We might then deduce IMH from X. In this case we know what we are talking about — that is, we know the subject matter — but we merely “get out as much as we put in”. Not so interesting. (ii) We can be vague about what we mean: For example, we can say that we are talking about countable transitive models of ZFC that are “maximal” (with respect to something). But in that case we have little idea of what we are talking about (our subject matter) and it seems that “anything goes”.

You seem to want to resolve the conflict in (a) — between the claim
that inaccessibles are intrinsically justified and the claim that IMH
is intrinsically justified — by resorting to both intrinsic justifications on the basis of our concept of set (which gives inaccessibles) and intrinsic justifications on the basis of the hyperuniverse (understood as either (i) or (ii) under (c)) and which gives IMH) and you seem to want to leverage the interplay between these two in such a way that it gives us information about our concept of set (which concerns V). But what can you say about the relationship between these two forms of intrinsic justification? Is there some kind of “meta” (or “cross-domain”) form of intrinsic justification that is supposed to give us confidence about why intrinsic justifications on the basis of the hyperuniverse should be accurate indicators of truth (or intrinsic justifications on the basis of) our concept of set?

One final comment: Here is an “intuition pump” regarding the claim that IMH is intrinsically justified.

THE FACTS:

1. If there is a Woodin cardinal with an inaccessible above then IMH
is consistent.
2. If IMH holds then measurable cardinals are consistent.

So, if IMH is intrinsically justified (in the standard sense) then we can lean on it to ground our confidence in the consistency of measurable cardinals. For my part, the epistemic grounding runs the other way: IMH provides me with no confidence in the consistency of measurable cardinals (or of anything). Instead, the consistency of IMH is something in need of grounding. Fact (1) above provides me with evidence that IMH is consistent. Fact (2) does not provide me with evidence that measurable cardinals are consistent. I think most would agree. If I am correct about this then it raises further problems for the claim that IMH is intrinsically justified (in the standard sense).

I have further comments and questions about your notion of “sharp-generated reflection” and how you use it to modify IMH to $\textsf{IMH}^\#$. But those questions seem premature at this point, given that I am not on board with the basics. Let me just say this: The fact that you are readily modify (intrinsically justified) IMH to $\textsf{IMH}^\#$ in light of the fact that IMH is incompatible with (intrinsically justified) inaccessibles indicates that your notion of intrinsic justification is quite revisable and, I think, best regarded as “intrinsic plausibility” or “extrinsic justification” or something else

Best,
Peter